A Calabi–Yau manifold of complex dimension is a smooth compact complex manifold, most often a smooth projective variety, with trivial canonical bundle. Equivalently its first Chern class vanishes, , equivalently it carries a nowhere-vanishing holomorphic -form.
The metric side follows from this algebraic condition. By Yau’s theorem, which proves the Calabi conjecture, a compact Kähler manifold with admits a unique Ricci-flat Kähler metric in each Kähler class. This is equivalent to the holonomy group lying in .
Examples arrange by complex dimension: the elliptic curve at , the K3 surface at , and the quintic at .
The case has six real dimensions, matching the number of hidden spatial directions in superstring theory, which is why it governs string-theory compactification.
The quintic
The quintic threefold is the standard running example of a compact Calabi–Yau threefold. It is the smooth hypersurface cut out by a degree-five homogeneous polynomial in $\mathbb{CP}^4$, of complex dimension 3. The Fermat member is the maximally symmetric representative, carrying the symmetry group up to permutations of coordinates.
Triviality of its canonical bundle follows by adjunction, since the canonical bundle of a degree- hypersurface in is , which at and is .
Its Hodge numbers are and , and it is the classic worked example for mirror symmetry and curve counting.